The relation between ray-trace and diffraction tomography is usually obscured by formulation of the two methods in different domains: the former in space, the latter in wavenumber. Here diffraction tomography is reformulated in the space domain, under the title of wave-equation tomography. With this transformation, wave-equation tomography projects monochromatic, scattered wavefields back over source-receiver wavepaths, just as ray-trace tomography projects traveltime delays back over source-receiver raypaths. Derived under the Born approximation, these wavepaths are wave-theoretic backprojection patterns for reflected energy; derived under the Rytov approximation, they are wave-theoretic back-projection patterns for transmitted energy.Differences between ray-trace and wave-equation tomography are examined through comparison of wavepaths and raypaths, followed by their application to a transmission-geometry, synthetic data set. Rytov wave-equation tomography proves superior to ray-trace tomography in dealing with geometrical frequency dispersion and finite-aperture data, but inferior in robustness. Where ray-trace tomography assumes linear phase delay and inverts the arrival time of one well-understood event, wave-equation tomography accommodates scattering and inverts all of the signal and noise on an infinite trace simultaneously. Interpreted through the uncertainty relation, these differences lead to a redefinition of Rytov wavepaths as monochromatic raypaths, and of raypaths as infinite-bandwidth wavepaths (Rytov wavepaths averaged over an infinite bandwidth).The infinite-bandwidth and infinite-time assumptions of ray-trace and Rytov, wave-equation tomography are reconciled through the introduction of bandlimited raypaths (Rytov wavepaths averaged over a finite bandwidth). A compromise between rays and waves, bandlimited raypaths are broad backprojection patterns that account for the uncertainty inherent in picking traveltimes from bandlimited data.

Wave-equation tomography

The authors explore a computational method for reconstructing an n-dimensional signal f from a sampled version of its Fourier transform f/spl circ/. The method involves a window function w/spl circ/ and proceeds in three steps. First, the convolution g/spl circ/=w/spl circ/*f/spl circ/ is computed numerically on a Cartesian grid, using the available samples of f/spl circ/. Then, g=wf is computed via the inverse discrete Fourier transform, and finally f is obtained as g/w. Due to the smoothing effect of the convolution, evaluating w/spl circ/*f/spl circ/ is much less error prone than merely interpolating f/spl circ/. The method was originally devised for image reconstruction in radio astronomy, but is actually applicable to a broad range of reconstructive imaging methods, including magnetic resonance imaging and computed tomography. In particular, it provides a fast and accurate alternative to the filtered backprojection. The basic method has several variants with other applications, such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the Radon (Hough) transform. >

The gridding method for image reconstruction by Fourier transformation

Tomographic imaging is now in widespread use in geophysical inversion. Most early work in this field used the ray approximation to wave propagation, but more recently scattering effects have been addressed via the formalism of diffraction tomography. However, if the correct image cannot be adequately represented as a perturbation of a simple background, this may demand considerably more computational resources than ray tomography. Therefore it is of some interest to determine the scalelength of variation below which scattering cannot be neglected, so that ray tomography is no longer reliable. In the hypothetical case of monochromatic illumination of a region consisting of small perturbations to a homogeneous background, the way in which the two reconstruction algorithms map the data into the Fourier components of the object may be directly compared to reveal the effect of neglecting scattering. These mappings begin to differ significantly at a wave‐number corresponding to the first Fresnel zone radius; it...

A guide to the limits of resolution imposed by scattering in ray tomography

A unified theoretical framework based on the scalar wave equation is presented for transmission tomography of weakly inhomogeneous objects embedded in a known uniform background medium The developed formulation differs from the usual treatments of diffraction tomography in that it is not limited to applications employing plane-wave illumination and planar measurement boundaries and can be extended to weakly inhomogeneous objects embedded in non-uniform backgrounds and to the class of strongly scattering objects The new formulation is based on the mathematical operations of propagation and backpropagation which are generalisations of the conventional tomographic operations of projection and backprojection to the case of diffracting wavefields The projection and backprojection operations are then obtained from the generalised operations in the zero wavelength limit A class of reconstruction algorithms are developed within the new formulation that are generalisations, to the case of diffraction tomography, of the class of iterative ART (algebraic reconstruction technique) algorithms of conventional (X-ray) tomography The reconstruction algorithms developed (called generalised ART algorithms) are shown to be applicable to situations where the tomographic data are sparse (limited-view problem) and to the incorporation of a priori information into the reconstruction process

Reconstructive tomography with diffracting wavefields

Ultrasonic scattering is related to tissue architecture by a model which expresses the scattered wave pressure as the product of a frequency-dependent factor and the three-dimensional Fourier transform of a spatially windowed scattering function. The scattering function is the sum of the variations in compressibility and the variations in density, the latter being weighted by the cosine of the scattering angle. An analogous Fourier transform relation describes the average scattered intensity in terms of the correlation of the scattering function between points in a scattering volume. These Fourier transforms permit different scattering measurements to be related via trajectories in wavespace. Backscatter measurements made on blood, eye, liver, spleen, brain, and heart show the feasibility of differentiating tissues and determining spacing. Ultrasonic measurements of random medium model properties have demonstrated that average differential scattering cross sections per unit volume can be found accurately and precisely, and that scattering cross sections per unit volume can be used to make quantitative statements about changes in scattering properties. Measurements of calf liver ultrasonic differential and total scattering cross sections show that calf liver is a weak scatterer with nearly one-half of the total scattered power occurring between scattering angles of 25-45°. Measurements of scattering by fixed pig and human liver show qualitative agreement with predictions of scattering from architecture observed optically. Results presently accumulated indicate the promise of scattering measurements for tissue characterization, but extensive additional in vitro and in vivo development is required before their clinical utility for diagnosis is known.

A Review of Tissue Characterization from Ultrasonic Scattering

On distingue 2 problemes dans le traitement du signal; le premier concerne la reconstruction d'une image a partir de donnees mesurees transformees de facon appropriee qui echantillonnent le plan de frequences spatiale le long d'arcs circulaires; le second probleme est celui de desenroulement de phase des profils mesures

M. Soumekh

Papers

Signal Processing for Diffraction Tomography